Global well-posedness of some high-order focusing semilinear evolution equations with exponential nonlinearity

AbstractIn even space dimensions, the initial value problems for some high-order focusing semilinear evolution equations with exponential nonlinearities are considered. Using the potential well method, global and nonglobal well-posedness in the energy space are obtained.

Download Full-text

ENERGY CRITICAL NLS IN TWO SPACE DIMENSIONS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891609001927 ◽

2009 ◽

Vol 06 (03) ◽

pp. 549-575 ◽

Cited By ~ 24

Author(s):

J. COLLIANDER ◽

S. IBRAHIM ◽

M. MAJDOUB ◽

N. MASMOUDI

Keyword(s):

Schrödinger Equation ◽

Initial Value Problem ◽

Schrodinger Equation ◽

Nonlinear Schrodinger Equation ◽

Energy Space ◽

Well Posedness ◽

Initial Value ◽

Exponential Nonlinearity ◽

Supercritical Case ◽

Two Space Dimensions

We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with exponential nonlinearity [Formula: see text] We identify subcritical, critical, and supercritical regimes in the energy space. We establish global well-posedness in the subcritical and critical regimes. Well-posedness fails to hold in the supercritical case.

Download Full-text

Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091516000213 ◽

2016 ◽

Vol 60 (2) ◽

pp. 481-497 ◽

Cited By ~ 3

Author(s):

Tarek Saanouni

Keyword(s):

Heat Equation ◽

Finite Time ◽

Blow Up ◽

Type Equation ◽

High Order ◽

Energy Space ◽

Well Posedness ◽

Exponential Nonlinearity

AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

Download Full-text

Remarks on the critical nonlinear high-order heat equation

Arab Journal of Mathematical Sciences ◽

10.1016/j.ajmsc.2019.03.002 ◽

2019 ◽

Vol 26 (1/2) ◽

pp. 127-152

Author(s):

Tarek Saanouni

Keyword(s):

Global Existence ◽

Heat Equation ◽

Initial Value Problem ◽

Existence Of Solutions ◽

High Order ◽

Well Posedness ◽

Potential Well ◽

Initial Value ◽

Global Existence Of Solutions ◽

Nonlinear High Order

The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

Download Full-text

Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽

10.3390/math8112023 ◽

2020 ◽

Vol 8 (11) ◽

pp. 2023

Author(s):

Christopher Nicholas Angstmann ◽

Byron Alexander Jacobs ◽

Bruce Ian Henry ◽

Zhuang Xu

Keyword(s):

Initial Value Problems ◽

Fractional Derivatives ◽

Evolution Equations ◽

Integral Transform ◽

Initial Value ◽

Intrinsic Nature ◽

Recent Interest ◽

Special Cases ◽

Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

Download Full-text